Multilinear Harmonic Analysis and Applications

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Harmonic analysis is concerned with decomposing signals or functions into elementary waves, analyzing the pieces individually, and then synthesizing this local information into information about the original object. This approach has wide applications in physics and engineering, but also within mathematics itself. A special focus of harmonic analysis is to obtain good quantitative information on the objects in question, which can often be applied to yield strong quantitative results in the fields of ergodic theory and combinatorics. Ergodic theory investigates behavior of dynamical systems that evolve for a long time, while one focus of combinatorics is to study existence of structures in large but otherwise arbitrary sets. These structures can be geometric patterns such as arithmetic progressions, vertices of a square, their translates, rotates, or dilates. The project will focus on the study of integral transformations that naturally occur within harmonic analysis and build a set of tools that further expands the reach of harmonic analysis into the nearby fields of ergodic theory and combinatorics. As these are very foundational subjects, the results and techniques from these areas are applicable in a wider scientific context. Examples include applications in stochastic analysis, partial differential equations, signal processing, and medical imaging. The project’s fundamental mathematical research will address the community’s currently relevant scientific questions in its continued pursuit in understanding open mathematical questions. The project will further entail working with a postdoc and mentoring undergraduate students, organization of seminars and workshops that promote cross-collaboration and exchange of ideas. The first part of this project will deal with a fundamental family of multilinear singular integral forms with a hypergraph structure. The main goal will be to address boundedness of these forms on Lebesgue spaces. These questions will be approached by extending the techniques previously developed by the PI and collaborators. The work is partially inspired by a major open question in harmonic analysis, boundedness of the simplex Hilbert transforms. Further, this project will address multilinear oscillatory integral forms and their connections with variants of multilinear singular integrals and maximal functions with curvature. Estimates on multilinear singular and oscillatory integrals will then be applied to ergodic theory. In this context, the main goal will be to investigate convergence of various types of ergodic averages along the orbits of commuting measure-preserving transformations by establishing qualitative convergence statements via stronger quantitative variational estimates. The latter will be translated to questions in Euclidean spaces and addressed with harmonic analysis techniques developed throughout this project. These methods will also be applied to questions in Euclidean Ramsey theory, where existence of linear and non-linear point configurations in large subsets of the Euclidean space will be investigated.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.